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On 4d rank-one N=3 superconformal field theories
Takahiro Nishinaka( Yukawa Institute )
July 26, 2016 @ Osaka U.
arXiv: 1602.01503 w/ Yuji Tachikawa ( IPMU )
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Introduction
4d SUSY QFTs
• N=1
• N=2
• N=4
SUSY standard model (MSSM)
exact computation (localization), SW curve
AdS/CFT, integrability
What about N=3 ???
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Introduction4d SUSY QFTs
( No genuine N=3 Lagrangian theory )
• N=3 Lagrangian always preserves N=4
• But these days, we have seen many non-Lagrangian QFTs.
e.g.) M5-branes 4d QFTcompactify
’16 [TN - Tachikawa]’15 [García-Etxebarria - Regalado]
’16 [Aharony - Tachikawa]
’15 [Aharony - Evtikhiev]
• There could be Non-Lagrangian genuine N=3 theories.
’16 [Imamura - Yokoyama]
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What do 4d genuin N=3 QFTs look like (if exist) ?Q:
Our answer
Focusing on 4d N=3 simplest conformal field theories ( SCFTs ),
• Moduli space of SUSY vacua
• conformal anomalies
• OPEs of BPS ops.
1/4 BPS operators described by 2d N=2 W-algebra
a = c =2` � 1
4
(` = 3, 4, 6)M = C3/Z`
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Outline
1. N=3 Lagrangian preserves N=4
2. N=3 (non-Lagrangian) SCFTs
3. The simplest examples
4. OPEs of BPS operators
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Outline
1. N=3 Lagrangian preserves N=4
2. N=3 (non-Lagrangian) SCFTs
3. The simplest examples
4. OPEs of BPS operators
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Supersymmetry
superchargeQ↵
Q†↵
boson fermion
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Aµ
�
N=3 Lagrangian preserves N=4
• massless N=1 SUSY multiplets
vector multiplet
1
1/2
0
-1
helicity
-1/2
1
1
�
chiral multiplet
1
2
1
vector chiral
CPT conjugate
1
1
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N=3 Lagrangian preserves N=4
1
1/2
0
-1
-1/2
vector hyper
1
2
2
1
2
• massless N=2 SUSY multipletshelicity
1
2
1hyper multiplet
2
4
2
q q
N=1 chiral
N=1 vector
N=1 chiralN=1 chiral
�1 �2
�
vector multiplet
Aµ
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N=3 Lagrangian preserves N=4
vector multiplet1
1/2
0
-1
-1/2
vector
1
4
4
1
6
• massless N=3 SUSY multiplets
This is equivalent to the N=4 vector multiplet!
N=2 vector N=2 hyper
helicity
3
1
1
3
The only N=3 massless multiplet is an N=4 vector multiplet!!
�1 �2
Aµ
�
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W = Tr �[Q, Q]
�
Q
Q
N=3 Lagrangian preserves N=4
• The only N=3 preserving renormalizable superpotential is
This is precisely the superpotential for N=4 SYM.
�1 �2
Aµ
What about interaction?
� q q
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In summary,
• The only massless N=3 multiplet is equivalent to an N=4 vector multiplet.
• Moreover, every N=3 (renormalizable) Lagrangian preserves N=4.
No genuine N=3 renormalizable Lagrangian!!
• This particularly means that every N=3 free SCFT preserves N=4 SUSY.
�1 �2
Aµ
�
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Outline
1. N=3 Lagrangian preserves N=4
2. N=3 (non-Lagrangian) SCFTs
3. The simplest examples
4. OPEs of BPS operators
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Let’s focus on 4d N=3 superconformal field theories.
Then, despite the lack of Lagrangian, we can learn about
( SCFT )( N=4 )
• Global symmetry
• Conformal anomalies
• Moduli space of SUSY vacua
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N=3 (non-Lagrangian) SCFTs
Global symmetry
• Bosonic global sym.
SO(4,2) x U(3)R x GF
• No N=3 flavor symmetry is allowed.
Jµ : flavor current for GF
J↵µ =⇥Q3
↵, Jµ
⇤
J↵µ =hQ3↵, Jµ
i extra supercurrents leading to N=4 SUSY
conformal N=3 R-sym N=3 flavor sym
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a =1
24, c =
1
12
N=3 (non-Lagrangian) SCFTs
Tµµ =
c
16⇡2(Weyl)2 +
a
16⇡2(Euler)
(Euler) = R2µ⌫⇢� � 4R2
µ⌫ + R2
(Weyl)2 = R2µ⌫⇢� � 2R2
µ⌫ +1
3R2
These coefficients capture the number of degrees of freedom in CFTs
e.g.) free hyper :
1
2
a
c
5
4• N=2 SUSY ’08 [ Hofman-Maldacena ]
Conformal anomalies
a
c= 1• N=3 SUSY ’15 [ Aharony, Evtikhiev ]
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M
N=3 (non-Lagrangian) SCFTs
conformal point
SUSY vacua
Moduli space of vacua
flat directions preserving N=3 SUSY=
( but breaking conformal sym. )
dimC M = 3 x integer
�, q, q 2
=
parameterized by the VEV of a scalar in a massless N=3 multiplet
=
N=3 vector multiplet
N=3 vector multiplet
� q q�1 �2
Aµ
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a = c
In summary, every genuine N=3 SCFT has
• Global symmetry
• Central charges
• Moduli space of SUSY vacua
conformal point
SUSY vacua3 x integerdimC M =
Tµµ =
c
16⇡2(Weyl)2 +
a
16⇡2(Euler)
SO(4,2) x U(3)R x GF
conformal N=3 R-sym
N=3 flavor sym
M
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Outline
1. N=3 Lagrangian preserves N=4
2. N=3 (non-Lagrangian) SCFTs
3. The simplest examples
4. OPEs of BPS operators
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conformal point
SUSY vacua
3 x integerdimC M =( rank )
The simplest examples are rank-one:
3dimC M =
We focus on this case and argue that
M = C3/Z`
(` = 3, 4, 6)
a = c =2` � 1
4
M
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4d rank-one N=3 SCFTs conformal point
SUSY vacua
M
is parameterized by the VEVs of 3 scalars in an N=3 vector multiplet.M
Moduli space of vacua
3dimC M =
MH
Mh�i hqi, hqi
( N=2 Coulomb branch ) ( N=2 Higgs branch )⊂ ,
N=3 R-sym.
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3 (q, q)
4d rank-one N=3 SCFTs
Higgs branch
conformal point
SUSY vacua
M
MH
U(3)R U(1)R x SU(2)R x U(1)F⊂
N=2 R-sym. N=2 flavor sym.
N=3 R-sym :
MH = C2/Z` ` = 3, 4, 5, 6, · · ·
MH is acted on by SU(2)R and U(1)F•
MH : hyper Kahler cone w/ U(1) isometry=)
charged
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C3/Z`
= C2/Z`C/Z`
4d rank-one N=3 SCFTsCoulomb branch
` = 3, 4, 5, 6, · · ·
N=3 R-sym.
Mh�i hqi, hqi
( N=2 Coulomb branch ) ( N=2 Higgs branch )⊂ +
=) ` = 3, 4, 6
e2⇡i`
h�i
h�i = 0
conformal point
The Seiberg-Witten curve is a singular -fiber over C/Z`T 2
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conformal point
SUSY vacua
M
(` = 3, 4, 6)M = C3/Z`
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2a � c =1
4
X
i
(2[�i] � 1)
4d rank-one N=3 SCFTsCentral charges
’08 [Shapere - Tachikawa]
Formula widely applicable to N=2 SCFTs
a = c
a = c =2` � 1
4
Combined w/ , we find
Scaling dim. of Coulomb branch operators
2a � c =1
4(2[�] � 1) =
2` � 1
4
In our case, read off from the SW curve
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In summary,
4d rank-one N=3 SCFTs should have...
• Moduli space of vacua
• Central charges
a = c =2` � 1
4
(` = 3, 4, 6)M = C3/Z`
` = 1
` = 2
c.f.) N=4 U(1) SYM
N=4 SU(2) SYM
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Outline
1. N=3 Lagrangian preserves N=4
2. N=3 (non-Lagrangian) SCFTs
3. The simplest examples
4. OPEs of BPS operators
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O1(x)O2(0) =X
k
c12k(x)Ok(0)
OPEs of 4d CFTs
• OPEs relate n-pt functions to (n-1)-pt functions.
Operator Product Expansion
• How to compute this without Lagrangian...?
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O4d(x) O2d(z)
’13 [Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees]
For any 4d N=2 SCFT
1/4 BPS operators 2d local operator
x
3,4
zz = x
1 + ix
2
R4 = R2 ⇥ R2
2
4d 2d map
O4d1 (x)O4d
2 (0) =X
k
c12k(x)O4dk (0) O2d
1 (z)O2d2 (0) =
X
k
O2dk (0)
zh1+h2�hk
4d OPE 2d chiral algebra
The detail of the 2d chiral algebra depends on the 4d SCFT you are studying.
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What are the 2d chiral algebras corresponding to the 4d N=3 SCFTs we are studying?
Q:
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J
OPEs of BPS operators
SU(2)R current N=2 super Virasoro
4d 2d
WfW
�C2/Z`
�Higgs branch operators
W fW ⇠ J `Higgs branch relation :
chiral primary
anti-chiral primary
• 4d N=3 SUSY implies 2d N=2 SUSY
c2d = �12c4d = �3(2` � 1)w/
(` = 3, 4, 6)
The label of 4d N=3 SCFT
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OPEs of BPS operators
4d
J
↵ ↵
�+↵ ��
↵
W+ W�
J i↵↵
stress tensor multiplet Higgs branch operator multiplet
J` = W+W�
G GW W
H H
N=2 super Virasoro
T
J
chiral primary anti-chiral primary
2d
stress tensor
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h =`
2
Our conjecture
• We assume all the 2d operators are generated by these.
• The 2d chiral algebra associated w/ the 4d rank-one N=3 SCFT for is an N=2 W-algebra generated by
J
W
fW
c2d = �3(2` � 1)N=2 super Virasoro
+chiral primary
anti-chiral primary+ w/
w/
[Odake]
[Inami-Matsuo-Yamanaka]
[Blumenhagen]
` = 3, 4, 6
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W(Z)W(0) fW(Z) fW(0)
W(Z) fW(0)
Ansatz for OPEs
~ regular ~ regular
~ composites of and its descendantsJ
Requirement from Jacobi identities
The above ansatz should be consistent with Jacobi identities
c2dconstraint on the value of
null operator relations
OPEs of BPS operators
chiral primary anti-chiral primary
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Requirement from Jacobi
`
�15
�21
�3318, �15,
12, �9,
c2dallowed values of
3
4
6
c2d = �12c4d
= �3(2` � 1)
expected value from 4d
W fW ⇠ J `
4d Higgs branch relation is realized as a null relation
( Only for these values, )
OPEs of BPS operators
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M = C3/Z` a = c =2` � 1
4
Summary• We have studied 4d rank-one N=3 SCFTs.
` = 3, 4, 6• The theories are labeled by .
• The corresponding 2d chiral algebras are conjectured to be N=2 W-algebras.
• There is no Lagrangian genuine N=3 theory. Therefore they have to be non-Lagrangian.
moduli space of SUSY vacua central charge
Open problemThe superconformal index of these N=3 SCFTs
The character of the 2d W-algebra we identified
=